This proposition says that if b is a smaller number than a, then b is either a part of a, that is, b is a unit fraction of a, or b is parts of a, that is, a proper fraction, but not a unit fraction, of a. For instance, 2 is one part of 6, namely, one third part; but 4 is parts of 6, namely, 2 third parts of 6.

It seems obvious that when one number b is less than another a, that since the unit u is a part of a and b is a multiple of u, then b is some multiple of a part of a. Yet, the proof of this proposition ignores that possibility, except in the special case when b and a are relatively prime. In the case of a = 4u and b = 6u, the proof will find that a is 2 third parts of b. Thus, it appears that a satisfactory answer to the question “How mary parts of a is b?” requires finding the least number of parts.

The proof has three cases.

If b and a are relatively prime, then b consists of b of the nth parts of a where a = nu.

If b divides a, then b is one part of a.

Otherwise they’re not relatively prime, and b does not divide a. Let d be their greatest common divisor. Then b is some multiple m of d, that is, b = md, and a is some other multiple n of d, that is, a = nd Therefore, b consists of m one-nth parts of a.

Again, if 1 were considered a number, then the three cases could be consolidated into one.

We'll write the statement such as b is one nth part of a as the equation b = a/n, and a statement such as b consists of m one-nth parts of a as the equation b = (m/n)a.